How to Solve a Math Brain Teaser and Land Your Dream Job

Did you figure out the answer to the brain teaser that I gave as an example of a question given during Google or Microsoft job interviews? Now you can find out if you were correct.

Jason Marshall, PhD,
October 12, 2011

How to Solve a Math Brain Teaser and Land Your Dream Job

How many points are there on a sphere where after walking one mile south, then one mile east, and finally one mile north, you end up right back where you started?

If you listened to or read the most recent Math Dude article, you might remember this brain teaser question. You might also remember that it’s an actual question that job seekers at Google and Microsoft have been asked in interviews. As such, if you hope to land a job at a company like that, you’d best learn how to think on your feet so that you can prove you have the chops to make it in that demanding world.

So how do you solve this puzzle? Well, first of all, if you haven’t yet taken some time to think about how to solve it, I encourage you to pause for a minute or two (or three or four) to do so. Then, when you’re ready for it, here’s my answer:

First, it’s not too tough to come up with one place on a sphere that solves the puzzle: the point right at the top. If you imagine standing at Earth’s North Pole, you’ll find that any direction you walk is south. So walking one mile south (which means any direction), then one mile east, then one mile north lands you right back where you started from at the North Pole.

But is that it? Or are there perhaps other locations that work as well?

Indeed there are other locations. In fact, there are an infinite number of them! How is that possible? Well, think about this: Somewhere near Earth’s South Pole, there’s a latitude that has a circumference of one mile. In other words, if you’re at this latitude and start walking east or west, in one mile you’ll be right back where you started from. If you instead start at some point one mile north of this latitude, your trip will take you one mile south to this special latitude, then one mile east “around the globe,” and finally one mile north right back to wherever you started from. And it doesn’t matter precisely where you started on the line around the globe one mile north of this special latitude…there are an infinite number of possibilities that all work.

But we’re still not finished. If you think about it, you’ll see that the latitudes near the South Pole that have circumferences of 1/2 mile, 1/3 mile, 1/4 mile, and so on will always bring you right back to the point where you started from after several trips “around the world.” Which means that there are even infinitely more points that happily solve our puzzle.

You’ll find that brain teaser puzzles often have solutions like this in which one answer desperately tries to tempt you into believing that it’s the one and only solution. So if you feel that urge, be cautious. As with most things in life, spending time thinking about problems almost always opens doors to other possibilities.

Image via woodleywonderworks/Flickr

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